After that nasty diversion into economics and politics, we now return to your
\nregularly scheduled math blogging. And what a relief! In celebration, today I’ll give
\nyou something short, sweet, and beautiful: quotient groups. To me, this is a shining
\nexample of the beauty of abstract algebra. We’ve abstracted away from numbers to these
\ncrazy group things, and one reward is that we can see what division really means. It’s
\nmore than just a simple bit of arithmetic: division is a way of describing a fundamental
\nstructural relationship that pervades mathematics.<\/p>\n
<\/p>\n
So what is division all about?<\/p>\n
Suppose you want to divide 50 by 5. What you’re really doing is saying
\nyou’ve got a collection of 50 indistinguishable things, and you want to break it into a 5
\nindistinguishable collections. Since you started with 50 indistinguishable things, that means
\nthat you’ll end up with 5 sets of 10 things.<\/p>\n
That’s pretty simple, right? Now, suppose that we’re not talking about simple numbers. Instead we
\nwant to work in terms of groups. Can we take that basic concept of division, and find some meaningful way of applying it to groups?<\/p>\n
Well, first, we need to somehow talk about division in a way that doesn’t involve numbers. We start with a group – that is, a collection of objects with some kind of meaningful structure. What can we divide it by? A group with a similar structure – in fact, a group with the same structure: a subgroup But not just any subgroup: it’s got to be a normal<\/em> subgroup, because that’s the kind of subgroup that properly preserves the structure of the group.<\/p>\n As a reminder, a normal<\/em> subgroup (N,+) of a group (G,+) is a subgroup where ∀n∈N, ∀g∈G, g+n+g-1<\/sup>∈N. That is, it’s a normal subgroup group So what happens when we divide a group by one of its normal subgroups? We partition<\/em> the To be precise, given a group (G,×), and a normal subgroup (N,×), the members of So why the restriction for quotients to only be defined for normal subgroups? It’s pretty A beautiful example of the quotient group comes from the integers. Take the group of After that nasty diversion into economics and politics, we now return to your regularly scheduled math blogging. And what a relief! In celebration, today I’ll give you something short, sweet, and beautiful: quotient groups. To me, this is a shining example of the beauty of abstract algebra. We’ve abstracted away from numbers to these crazy […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[26],"tags":[],"class_list":["post-569","post","type-post","status-publish","format-standard","hentry","category-group-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9b","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/569","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=569"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/569\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=569"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nwhich is closed with respect to the group operation performed in sequence with any member of the group and that members inverse.<\/p>\n
\ngroup into a new group, where the elements of the new group are formed from subsets<\/em> of the
\nelements of the original group. It’s the same idea as simple integer division described up above, except that we want to preserve the group structure, so the result is going to be a group.<\/p>\n
\nthe quotient group G\/N are the set of set products of elements of G and the set N:
\n∪g∈G<\/sub>{ { g×n | n∈N } }, which can also be written
\n{g×N | g∈G}. That is, each member of G\/N is the set of products of a member of G with each of the members of N. For a given member of the quotient group, g×N, the inverse element
\nis g-1<\/sup>×N. The group operation of G\/N is set-product, and the identity element of G\/N is the set containing the identity element of G.<\/p>\n
\nsimple: the construction above will only<\/em> be a group if N is normal. Note the way we multiply a member of the group by the members of the subgroup? If we work that out, the only way
\nthat the quotient group is closed under the group operation of set-product is if the subgroup
\nthat generated the quotient is normal.<\/p>\n
\nintegers with addition as the operator: (Z<\/b>,+). The set of even numbers, 2Z<\/b> is
\na normal subgroup of the integer group. The quotient group Z<\/b>\/2Z<\/b> is
\na two element group: one element is the set of all even integers; the other is the set of all
\nodd integers. This is equivalent to the cyclic group of size two – aka, mod-2 arithmetic. You can do a similar trick with the reals and the integers, where the R<\/b>\/Z<\/b> gives you a sort of real-valued modulo group formed from the set of all reals between 0 and 1.<\/p>\n","protected":false},"excerpt":{"rendered":"